#8602762

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(17,18,22,24)(19,23,27,28)(20,25,30,32)(21,26,31,29)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(2,7)(3,12)(4,10)(14,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(34,37)(40,43)(41,45)(47,48);(2,7)(4,10)(8,11)(13,16)(18,24)(19,27)(25,32)(26,29)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20,22,30)(18,25,24,32)(19,26,27,29)(21,23,31,28)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,31)(30,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21)(18,26)(19,25)(20,28)(22,31)(23,30)(24,29)(27,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509958);